Skew Constacyclic Codes Of Length $np^s$ over $ \frac{\mathbb{F}_{p^m}[u]}{\langle u^k \rangle}

Abstract: Let $\mathbb{F}<em>{p<sup>m}$ be the field containing $p<sup>m$ elements where $p$ is an odd prime and $m \in \mathbb{N}$. In this article, we propose a unified approach to the study of skew constacyclic codes of length $np<sup>s$ over the ring $$R_k = \mathbb{F}</em>{p^{m}[u]/\langle} u^k \rangle,$$ where $n, s, k \in \mathbb{N}$ and $\gcd(n, p)=1$. Consider the skew polynomial ring $R_k[x;Θ]$, where $Θ$ is an automorphism of $R_k$ such that $xa = Θ(a)x$ for all $a \in R_k$. Let $f(x)$ be a central irreducible divisor of $x^{{np^s}} - λ$ of degree $l$ and multiplicity $j$ in $R_k[x;Θ]$, where $λ$ is an invertible element in $R_k$. In this article, we study skew constacyclic codes of length (np^s) over (R_k), which reduces to the study of skew polycyclic codes of length $jl$ associated with a polynomial (f(x)^j). Using the fact that skew polycyclic codes associated with a polynomial (f(x)^j) can be described by the left ideal structure of the quotient ring $R_k[x;Θ]/\langle f(x)<sup>{j}\rangle$, we investigate this class of codes for specific choices of $Θ$. In particular, if $λ$ is an invertible element of $\mathbb{F}_{p<sup>m}$, we classify all left ideals and establish an isomorphism between skew cyclic and skew constacyclic codes, under suitable conditions. Furthermore, we provide a comprehensive analysis of skew constacyclic codes of length $3p^s$ over $R_k$. Finally, we examine skew cyclic and skew negacyclic codes of length $6p^s$ over $R_k$ using the factorization of $x^{{6p^s}} - 1$ and $x^{{6p^s}} + 1$, respectively; with a complete case-by-case analysis. Examples demonstrating codes with optimal parameters are also included.